The Furstenberg multiple recurrence theorem can be phrased as follows: if is a probability space with a measure-preserving shift (which naturally induces an isomorphism by setting ), is non-negative with positive trace , and is an integer, then one has

In particular, for all in a set of positive upper density. This result is famously equivalent to Szemerédi’s theorem on arithmetic progressions.

The Host-Kra multiple convergence theorem makes the related assertion that if , then the scalar averages

converge to a limit as ; a fortiori, the function averages

converge in (say) norm.

The space is a commutative example of a von Neumann algebra: an algebra of bounded linear operators on a complex Hilbert space which is closed under the weak operator topology, and under taking adjoints. Indeed, one can take to be , and identify each element of with the multiplier operator . The operation is then a finite trace for this algebra, i.e. a linear map from the algebra to the scalars such that , , and , with equality iff . The shift is then an automorphism of this algebra (preserving shift and conjugation).

We can generalise this situation to the noncommutative setting. Define a von Neumann dynamical system to be a von Neumann algebra with a finite trace and an automorphism . In addition to the commutative examples generated by measure-preserving systems, we give three other examples here:

(Matrices) is the algebra of complex matrices, with trace and shift , where is a fixed unitary matrix.

(Group algebras) is the closure of the group algebra of a discrete group (i.e. the algebra of finite formal complex combinations of group elements), which acts on the Hilbert space by convolution (identifying each group element with its Kronecker delta function). A trace is given by , where is the Kronecker delta at the identity. Any automorphism of the group induces a shift .

(Noncommutative torus) is the von Neumann algebra acting on generated by the multiplier operator and the shifted multiplier operator , where is fixed. A trace is given by , where is the constant function.

Inspired by noncommutative generalisations of other results in commutative analysis, one can then ask the following questions, for a fixed and for a fixed von Neumann dynamical system :

(Recurrence on average) Whenever is non-negative with positive trace, is it true that

(Recurrence on a dense set) Whenever is non-negative with positive trace, is it true thatfor all in a set of positive upper density?

(Weak convergence) With , is it true thatconverges?

(Strong convergence) With , is it true thatconverges in using the Hilbert-Schmidt norm ?

Note that strong convergence automatically implies weak convergence, and recurrence on average automatically implies recurrence on a dense set.

For , all four questions can trivially be answered “yes”. For , the answer to the above four questions is also “yes”, thanks to the von Neumann ergodic theorem for unitary operators. For , we were able to establish a positive answer to the “recurrence on a dense set”, “weak convergence”, and “strong convergence” results assuming that is ergodic. For general , we have a positive answer to all four questions under the assumption that is asymptotically abelian, which roughly speaking means that the commutators converges to zero (in an appropriate weak sense) as . Both of these proofs adapt the usual ergodic theory arguments; the latter result generalises some earlier work of Niculescu-Stroh-Zsido, Duvenhage, and Beyers-Duvenhage-Stroh. For the result, a key observation is that the van der Corput lemma can be used to control triple averages without requiring any commutativity; the “generalised von Neumann” trick of using multiple applications of the van der Corput trick to control higher averages, however, relies much more strongly on commutativity.

In most other situations we have counterexamples to all of these questions. In particular:

For , recurrence on average can fail on an ergodic system; indeed, one can even make the average negative. This example is ultimately based on a Behrend example construction and a von Neumann algebra construction known as the crossed product.

For , recurrence on a dense set can also fail if the ergodicity hypothesis is dropped. This also uses the Behrend example and the crossed product construction.

For , weak and strong convergence can fail even assuming ergodicity. This uses a group theoretic construction, which amusingly was inspired by Grothendieck’s interpretation of a group as a sheaf of flat connections, which I blogged about recently, and which I will discuss below the fold.

For , recurrence on a dense set fails even with the ergodicity hypothesis. This uses a fancier version of the Behrend example due to Ruzsa in this paper of Bergelson, Host, and Kra. This example only applies for ; we do not know for whether recurrence on a dense set holds for ergodic systems.

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